Geometrical frustration yields fiber formation in self-assembly

TL;DR

This paper demonstrates that geometrical frustration in irregular particles with short-range interactions naturally leads to the formation of fibers during self-assembly, providing insights into biological and synthetic fiber formation.

Contribution

It introduces a minimal model showing how geometrical frustration causes fiber formation in self-assembling particles, regardless of particle shape.

Findings

Fibers form robustly across various particle shapes and conditions.

Geometrical frustration determines the parameter range for fiber formation.

Fibers exhibit metastable behavior influenced by particle irregularity.

Abstract

Controlling the self-assembly of supramolecular structures is vital for living cells, and a central challenge for engineering at the nano- and microscales. Nevertheless, even particles without optimized shapes can robustly form well-defined morphologies. This is the case in numerous medical conditions where normally soluble proteins aggregate into fibers. Beyond the diversity of molecular mechanisms involved, we propose that fibers generically arise from the aggregation of irregular particles with short-range interactions. Using a minimal model of ill-fitting, sticky particles, we demonstrate robust fiber formation for a variety of particle shapes and aggregation conditions. Geometrical frustration plays a crucial role in this process, and accounts for the range of parameters in which fibers form as well as for their metastable character.

Equations

$$e^{(\alpha )}=\frac{1}{A^{(\alpha )}}+\sum _{i=1}^n\frac{k_i}{2}{\left[{\ell }_i^{(\alpha )}\right]}^2,$$

$$E(\{k_i\},\sigma ,\mathcal{T})=\sum _{\alpha =1}^N\left[e_{\mathcal{T}}^{(\alpha )}-e_g\right]+N_u(\mathcal{T})\sigma .$$

$$e_g=2\sqrt{2}{\left(5-2\sqrt{5}\right)}^{1/4},e_b=\sqrt{6},$$

$$e_g=2\sqrt{2(\sqrt{2}-1)},e_b=\frac{2^{3/2}}{5^{1/4}},$$

$$e_g$$

$$e_b$$

Full text

Geometrical frustration yields fiber formation in self-assembly

Martin Lenz

martin.lenz@u-psud.fr

LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405 Orsay, France

  

Thomas A. Witten

Department of Physics and James Franck Institute, University of Chicago, Chicago, Illinois 60637, United States.

(March 18, 2024)

**Controlling the self-assembly of supramolecular structures is vital for living cells, and a central challenge for engineering at the nano- and microscales Glotzer and Solomon (2007); McManus et al. (2016). Nevertheless, even particles without optimized shapes can robustly form well-defined morphologies. This is the case in numerous medical conditions where normally soluble proteins aggregate into fibers Eaton and Hofrichter (1990); Knowles et al. (2014). Beyond the diversity of molecular mechanisms involved Nelson and Eisenberg (2006); Eichner and Radford (2011), we propose that fibers generically arise from the aggregation of irregular particles with short-range interactions. Using a minimal model of ill-fitting, sticky particles, we demonstrate robust fiber formation for a variety of particle shapes and aggregation conditions. Geometrical frustration plays a crucial role in this process, and accounts for the range of parameters in which fibers form as well as for their metastable character. **

Identical cubes can pack into dense space-filling aggregates, but most shapes do not. As a result, the aggregates formed by these shapes tend to be frustrated, giving rise to arrested, glassy states Foffi et al. (2002); Cardinaux et al. (2007). In protein aggregates, this frustration can arise from, e.g., deformed or partially denatured protein domains, the juxtaposition of residues with unfavorable interactions or sterically hindered hydrogen bonding. Any compact packing of these objects thus involves tradeoffs between geometrical constraints, which hinder the formation of compact aggregates, and the particles’ overall attractive interactions. As a result, the global morphology of the aggregate is controlled by the competition between these two effects.

To explore this competition in its simplest form, we consider two-dimensional, deformable polygons driven to aggregate by zero-range attractive interactions [Fig. 1(a-b)]. We parametrize the magnitude of this attraction by a surface tension whose value controls the morphology of the aggregate [Fig. 1(c)]. A low surface tension thus favors thin tree-like aggregates composed of undeformed particles with very little elastic frustration, reminiscent of so-called empty liquids Bianchi et al. (2006). Conversely, a large surface tension leads to space-filling aggregates in which all particles are substantially deformed. In this paper, we demonstrate that fibers form at intermediate values of the tension, where the characteristic energies associated with particle attraction and deformation are comparable. We quantitatively account for these values based on the role of frustration, and show that fibers are very robust to changes in microscopic parameters, aggregation protocol and seeding conditions. Finally, we show that despite this robustness fibers do not constitute the ground state of our aggregates. Instead, they are kinetically trapped metastable states, consistent with their inherent frustration and with the well-documented irreversible character of protein fiber assembly in vivo.

We consider $n$-sided polygons, and use a deformation energy $e^{(\alpha)}$ for the $\alpha$th polygon that is a function of its area $A^{(\alpha)}$ and of the lengths $\{\ell_{i}^{(\alpha)}\}_{i=1\ldots n}$ of its sides as shown in Fig. 2(a). In the following we use both regular and irregular polygons [as in Fig. 1(b)], the latter being characterized by an asymmetry parameter $k\geq 1$, where $k\rightarrow 1$ is the regular polygon limit while $k\rightarrow+\infty$ yields short sides with vanishing spontaneous length (see Methods). Minimizing the energy $e^{(\alpha)}$ with respect to the positions of the polygon’s vertices yields a rigid elastic ground state of energy $e_{g}$ [Fig. 2(b)]. Aggregates are formed by connecting multiple polygons through the joining of one or several of their sides. Two joined sides are treated as a single object, implying that they share the same two end-vertices [Fig. 2(a)]. Side joining is favored by the adhesion energy between particles, modeled by an energy penalty $\sigma>0$ for each unjoined side regardless of its actual length [$\sigma$ thus parametrizes the surface tension introduced in Fig. 1(c)]. However, it also involves a distortion of the mismatched polygons, and thus increases their deformation energy above $e_{g}$.

A tree such as the one of Fig. 1(c) is always in its elastic ground state. Its average energy per particle is thus entirely due to surface tension, and reads $e\sim(n-2)\sigma$ in the thermodynamic limit (see Methods). A bulk, on the other hand, has a negligible surface energy in the thermodynamic limit but a finite deformation energy $e\sim e_{b}-e_{g}$, where $e_{b}$ denotes the minimal deformation energy for a polygon constrained by the bulk topology [Fig. 2(c)]. Rescaling the elastic energy and tension through $\tilde{e}^{(\alpha)}=[e^{(\alpha)}-e_{g}]/(e_{b}-e_{g})$ and $\tilde{\sigma}=(n-2)\sigma/(e_{b}-e_{g})$, we obtain $\tilde{e}=\tilde{\sigma}$ for a tree and $\tilde{e}=1$ for a bulk. This rescaled form for the energy makes it clear that adhesion overcomes frustrations and trees become less stable than bulks at high tensions, with a transition at $\tilde{\sigma}=1$. As a result, if fibers indeed form as a result of the competition between these two effects, we expect them to appear for a dimensionless tension of order one.

To test this hypothesis, we simulate irreversible aggregation starting from a single polygon. Our algorithm mimics irreversible protein aggregation, where a particle binding to an existing aggregate does so in the most energetically favorable location without substantial rearrangements of the preexisting aggregate topology. Throughout this process the aggregate energy is always minimized with respect to the position of all its vertices, implying that we impose mechanical force balance within an aggregate before assessing its energy (see Methods). We first grow aggregates of 150 irregular hexagons. The unfrustrated case $k=1$ simply yields the bulk of Fig. 1(a) up to very large values of $\sigma$. Next considering substantially frustrated hexagons with $k=2$, we observe bulks at high ($\tilde{\sigma}\geqslant 2.4$) tensions, while low tensions ($\tilde{\sigma}\leqslant 2.0$) yield irregular tree-like aggregates [Fig. 3(a)]. By contrast, periodic fibers form at intermediate tensions, and maintain perfect regularity to indefinitely large lengths (see Movie S1). These fibers form for $\tilde{\sigma}$ of order unity, consistent with our predicted competition between frustration and adhesion. Still, they form closer to $\tilde{\sigma}=2$ than the expected $\tilde{\sigma}=1$, suggesting that fiber formation is not completely captured by equilibrium arguments. To confirm this, we extrapolate the specific energy of our periodic fibers to infinite lengths and compare them to that of the hexagon bulk shown in Figs. 1(c) and 4(a). As shown in Fig. 4(b), the fiber energy exceeds that of the bulk, implying that fibers are indeed out of equilibrium.

To confirm that our fibers are metastable aggregates, we next establish that they are unaffected by small perturbations in the growth pathway but change morphology if nucleated from a more stable phase. To test the first point, we modify our algorithm to successively add two polygons, then remove one. Similar to polygon addition, our polygon removal procedure minimizes the aggregate energy in a short-sighted fashion, allowing the relaxation of built-up stresses and thus lowering the aggregate energy. The whole procedure is then iterated until an aggregate of the desired size is obtained. As expected, fibers are essentially unaffected by this local change in protocol [Fig. 3(b)]. We next grow an aggregate from a nucleus of the bulk, inducing significant morphological changes as predicted [Fig. 3(c)]. The unidirectional, periodic growth is however preserved, attesting to the robustness of the fiber-forming mechanism.

Moving beyond the $k=2$ hexagons considered above, Fig. 3(d-e) demonstrates that our description is valid for a broad range of $k$ corresponding to variations of the frustration energy $e_{b}-e_{g}$ by several orders of magnitude, from $\simeq 7\times 10^{-6}$ for $k=1.01$ to $\simeq 2\times 10^{-1}$ for $k=4$. By comparison the polygon ground state energy $e_{g}$ remains of order one over this whole range [Fig. 2(c)]. Despite these very substantial differences in the magnitude of the frustration energy, the rescaled parameter $\tilde{\sigma}$ remains an excellent predictor of fiber formation. Finally, we move away from hexagons altogether in Fig. 3(f-g) and demonstrate fiber formation in regular pentagons and octagons, two further shapes that do not tile the plane and thus generate intrinsically frustrated aggregates. Despite very diverse internal fiber structures, the onset of fiber formation is again very well predicted by the criterion $\tilde{\sigma}\simeq 2$.

Our results demonstrate that the inherent geometrical frustration of aggregates of mismatched particles gives rise to a richer range of morphologies than is found in well-matched objects. Most noticeable among these is the robust formation of fibers in regimes where particle adhesion and frustration are comparable in magnitude. Our fibers stand in strong contrast with the three-dimensional morphologies resulting from, e.g., the flocculation of simple spherical colloids. According to our analysis, the formation of slender aggregates is driven by a compromise between, on the one hand, the elastic incentive to place all particles in the vicinity of the boundary of the aggregate to relax their frustrated shapes, and on the other hand the tendency to form a compact aggregate that maximizes adhesion. Though our present demonstration of fiber formation is implemented in two dimensions, the frustration it reveals is also relevant in higher dimensions and will favor the formation of both fibers and sheets in 3D.

The currently dominant paradigm for frustration in soft matter systems assimilates incompatible shapes to a mismatch between an intrinsically curved Riemanian metric favored by the object and the flat metric of the embedding space Grason (2016). These concepts have been successfully applied to (quasi)crystals, surfactant phases or packings of preformed helical fibers, among others Sadoc and Mosseri (2008); Bruss and Grason (2012); Efrati et al. (2013). At equilibrium, this mismatch is accomodated by introducing defects in the system Bowick and Giomi (2009) or by forming slender morphologies if defects are strongly penalized Schneider and Gompper (2005); Hure et al. (2011); Meng et al. (2014); Hall et al. (2016); Sharon and Aharoni (2016). Here slender fibers and topological defects can indeed coexist when using highly frustrated polygons [pentagons, high-$k$ hexagons, octagons; see Fig. 4(c)]. However our fibers are distinctively out of equilibrium structures, and arise irrespective of whether the intrinsic Gaussian curvature of their constitutive polygons is positive (for pentagons), negative (octagons), or zero (irregular hexagons). These properties contrast with existing Riemanian metric models, suggesting a different frustration mechanism and posing the question of the interplay between the sculpting of the aggregate boundary and the formation of internal defects.

Turning to pathological fiber formation, our results suggest that the distinctive fibrous morphologies of protein aggregates need not be due to a mere coincidental convergence of the underlying molecular mechanisms, but could instead result from generic physical mechanisms. Indeed, while the formation of cross-$\beta$ spines is often discussed as the defining feature of one important class of such fibers, namely amyloids Knowles et al. (2014), deviations from this specific molecular organization have been observed Bousset et al. (2002) and secondary interactions have been shown to contribute significantly to their mechanics Knowles et al. (2007) and morphologies Meinhardt et al. (2009). These features are consistent with the diverse morphologies obtained in our model upon small variations of our parameters, and could apply to protein fibers with radically different structures Eaton and Hofrichter (1990). Beyond biological materials, fiber formation upon aggregation could become a hallmark of self-assembled, frustrated matter, leading to new design principles taking advantage of increasingly sophisticated artificial asymmetrical building blocks at the nano- and micro-scale Champion et al. (2007); Wang et al. (2012).